** LITERATURE REVIEW**

**2.3 Analytical Solutions for Saltwater Intrusion Ghyben-Herzberg Solution Ghyben-Herzberg Solution**

Analytical solution can be obtained based on the assumption that the groundwater flow system is in dynamic equilibrium between steady freshwater flow and static seawater, separated by a sharp interface. The Dupuit-Forchheimer approximation is adopted where the flow is predominantly horizontal, the vertical resistance to flow is neglected, and the pressure distribution is hydrostatic (Dupuit, 1863; Forchheimer, 1886). As shown in Figure 2.1, the thickness of the freshwater zone can be computed using the Ghyben-Herzberg formula (Badon Ghyben, 1889; Herzberg, 1901; Post, 2018): (dimensionless) is the relative difference between freshwater and seawater density, h (m) is the water table elevation (hydraulic head) above mean sea level (MSL), and ζ (m) is the depth of the freshwater-seawater interface below MSL.

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Figure 2.1: The Badon Ghyben-Herzberg principle: a freshwater-seawater interface in an unconfined coastal aquifer.

The average densities of freshwater and seawater are 1000 kg·m^{-3} and 1025 kg·m^{-3},
respectively, and thus *ζ is approximately equal to 40h and is always positive. This *
implies that a fall of the water table by 1 m will eventually lead to a rise of the
freshwater-seawater interface by 40 m. In the case of phreatic aquifers, the Dupuit
-Forchheimer approximation cannot account for the seepage face that may develop
above MSL, as illustrated in Figure 2.2 (Houben, 2015). The entire seepage area
potentially constitutes a vertical flow feature since the seepage boundary condition
allows discharge to take place, and thus the Dupuit assumption may predict a lower
water table in the vicinity of pumped wells.

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Figure 2.2: Schematic sketch of a seepage face in a well screened in an unconfined aquifer (Houben, 2015).

**Strack (1976) Analytical Solution **

Exact solutions for sharp-interface flow, in which the freshwater and seawater are
immiscible and separated by a sharp interface, can be obtained with the Strack’s
potential (Strack, 1976; Koussis et al., 2012). Conceptualization of simplified aquifer
settings is shown in Figure 2.3. The toe of seawater wedge, x*T* (m) represents a point
where the freshwater-seawater interface intersects the aquifer basement and
comprises a typical measure of the extent of saltwater intrusion. The aquifer domain
is separated into two zones, with freshwater flow in Zone 1 (inland of the interface, x

≥ x*T*) and interface flow in Zone 2 (bounded by the ocean boundary at *x = 0 and the *
toe location *x**T*). In the analytical solution of Strack (1976), the Dupuit-Forchheimer
assumption is applied to the freshwater flow, and the Ghyben-Herzberg relation is
utilized to define the interface depth. The freshwater head is constant along the
vertical in the freshwater zone and is a function of horizontal coordinates only, h*f* =
*h**f*(x, y), whereas flux is neglected in the saltwater zone. Using the discharge potential

22 Figure 2.3: Conceptualization of a steady-state sharp interface for unconfined aquifer

setting (Morgan et al., 2014).

Here, Ф (m^{2}) is the potential and *ζ*0 (m) is the depth of aquifer base below MSL.

These potential functions and their first derivatives are continuous across the
multiple zones of the aquifer and satisfy the Laplace equation in two horizontal
spatial dimensions, ∇^{2}Ф = 0, in the x-y plane (Felisa et al., 2013). Consider a vertical
cross-section of an unconfined aquifer with net recharge rate, *W *(m·s^{-1}), where *y is *
fixed, the governing equation is (Cheng and Ouazar, 1999):

###

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where K (m·s^{-1}) is the aquifer hydraulic conductivity. Boundary conditions are:

0 at 0, and *d* =*q*0 at 0,

*x* *x*

*dx* *K*

(2.5)

where *q**0* (m^{2}·s^{-1}) is the freshwater volume outflow rate per unit length of coastline.

The solution of Equation (2.4) subject to Equation (2.5) is

###

By substituting the potential value defined in Equation (2.6) into Equations (2.2) and (2.3), the hydraulic head, h (m) can be computed depending on the zone:

###

through substituting h = δ∙ζ0 in Equations (2.7) and (2.8) (Cheng and Ouazar, 1999).In situations where inflows from rainfall exceed the combined outflows from evapotranspiration and pumping, the net recharge rate is positive (W > 0). In saltwater intrusion cases, W value of either negative (total inflow < total outflow) or

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Various saltwater intrusion assessment methods have been developed based on the Strack (1976) analytical solution (Pool and Correra, 2011; Werner et al., 2012).

Werner et al. (2012) used the equations of Strack (1976) as the basis for developing
rapid assessment of saltwater intrusion vulnerability resulting from changes in sea
level, net recharge (W), and inflows at the inland boundary (q*b*). The representative
indicator of saltwater intrusion extent is the rate of change in the saltwater wedge toe
location in respond to changes in system stresses, as presented in Table 2.1. Werner
and Simmons (2009) reported that the impact of SLR in unconfined coastal aquifers
is smaller in flux-controlled systems (where groundwater discharge to the sea
remains constant) than in head-controlled systems (where groundwater hydraulic
head remains constant at the inland boundary). Flux control can be achieved by
adjusting the upstream groundwater management whereas head control can be
achieved through connection to a regulated surface water body, ensuring a certain
stage at that boundary.

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Table 2.1: Saltwater intrusion vulnerability indicator equations (Werner et al., 2012).

Flux-controlled setting Head-controlled setting

Sharp-interface analytical models work reasonably well for most large islands and coastal groundwater systems that have a relatively thin transition zone compared with the thickness of freshwater lens. However, atoll islands generally have thin freshwater lens overlying a much thicker transition zone, which cannot be realistically represented by a sharp-interface model (Lu et al., 2013). Such approximation may overestimate the extent of saltwater intrusion (Pool and Carrera, 2011), and thus underestimate the maximum allowable pumping rate. Several attempts have been made to realistically model the dispersive interface by taking into account the mixing of freshwater and seawater (Paster and Dagan, 2008). Henry (1964) developed a steady-state semi-analytical solution for saltwater encroachment in a confined coastal aquifer, including the effect of dispersion. The Henry’s solution has been used for benchmarking density-dependent flow models, but it has limited usefulness in simulating saltwater intrusion in real aquifers as only diffusion and no dispersion is simulated. Alternative solutions to the Henry problem were published later by Segol (1994), Simpson and Clement (2004), and Zidane (2012). Although the analytical models of density-dependent flow are more realistic than